Newton’s laws involve forces, and forces are vectors which
are a bit messier to handle and to think about than ordinary
functions are. In the Eighteenth and early Nineteenth Centuries
physicists got the idea of reformulating the laws of motion
in terms of energy functions particularly for systems of interacting
objects for which energy is conserved.
The most important such reformulation involves defining a
function called the Hamiltonian of the system. It is the
energy E that we have encountered above, but expressed not
in terms of position and velocity variables but rather in
terms of position and momentum variables.

For example, suppose we have a set of objects each with three
position variables and corresponding momentum variables. The
momentum variable pxi corresponding to xi
which itself is the x coordinate of the ith object,
is mivxi . The kinetic energy of the
object i is then .
If there is a potential energy of interaction between them
(such as that produced by gravitational attraction, there
will be a potential energy term of the form
between each pair of objects, i and j. The Hamiltonian, H,
of the system will then look like

The equations of motion, which correspond to F = ma
in this formulation are:
For each pi and ri, and each directiondwe have

(The subscript d here refers to directions x, y and z.),
These equations are called Hamilton’s equations.

In actuality they have the same content as Newton’s equations
in this context. Their importance lies particularly in that
quantum mechanics can be described most easily in terms of
the Hamiltonian.

If we choose a function Z of the position and momentum variables
here its time dependence can be computed by the chain rule
as

Substituting Hamilton’s equations here we get

The somewhat ugly last two terms here are called “the
Poisson Bracket” of Z and H, and written as {Z, H}, so
that we have

Exercises:

16.1 Consider the system consisting of the sun and the
earth, with a potential energy between them of .
Write down Hamilton’s equations for this system.

16.2 A force in the radial direction (plus or minus) is
called a central force. The force on the earth implied by
the example above is an example of one, if we choose the position
of the sun as origin. Compute the time derivative of reve
in this system for this (or any) central force.